Measures of Association

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Measures of Association
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Class Objectives

• Objectives Upon completion of this lesson the
  student will be able to
• Identify the appropriate measures of association
  to use for each scale of measurement.
• Based upon a statistical printout, describe the
  level of association between variables.
• Write the results in a format acceptable for a
  journal article.

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Relationship -- Association

• There is a relationship (or association) between
  variables when knowledge of one property
  (characteristic) of a case reduces uncertainty
  about another property (characteristic) of the
  case. A relationship (association) between
  variables means that variables tend to go
  together in a systematic way.

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Relationship Example
Percentage Point Increase
Hours of Studying Statistics
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Nominal Scale
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2 x 2 Contingency Table
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Phi Coefficient (?)

• Nominal Variables
• 2 x 2 Contingency Table
• A phi coefficient of zero indicates independence
  (no association) between variables
• A phi coefficient of /- 1 indicates complete
  dependence (association) between variables

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R x C Contingency Table
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Cramers V (?)

• Nominal variables
• Cramers V lies between 0 (complete independence)
  and 1 (complete dependence or association)
• Way to describe the apparent strength of
  statistical association in a contingency table
• Hard to put meaning in common sense terms,
  particularly if table has several rows and columns

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Measures of Relationships
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Ordinal Scale
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Ordinal Variables

• Use information about the ordering of categories
  by considering every possible pair of cases in
  the table
• Each pair of cases is checked to see if their
  relative ordering on the first variable is the
  same or reversed as their relative ordering on
  the second variable
• If the cases are ordered the same way on both
  variables, a positive relationship exists
• If the cases are ordered the differently on the
  variables, a negative relationship exists
• If no pattern exists, the variables are
  independent

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Square Contingency Table
Grade in AGEE 692
Number of Math Classes
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Kendalls tau b

• Appropriate for square contingency tables
• The value of tau will be between 1.0 and 1.0
• When there is no association, tau equals zero
• Can be interpreted as a difference between two
  proportions

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Kendalls tau c

• Appropriate for rectangular contingency tables
• Can attain /- 1.0 even if the two variables do
  not have the same number of categories.

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Spearman Rank-Order

• Variables are ranked
• Question How much is the ranking on variable x
  tend to agree with the ranking on variable y
• Spearman is a descriptive index of agreement
  between ranks

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What if?

• Nominal and Ordinal Variables
• Rank-biseral correlation coefficient

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Measures of Relationships
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Interval Scale
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Pearson Product-Moment Correlation Coefficient

• Absolute value of coefficient indicates magnitude
  of relationship
• Sign ( or -) indicates direction of
  relationship
• r designates coefficient from sample
• R (rho) designates coefficient for population

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Pearson Product-Moment

• Statistic r is not an unbiased estimate of
• Amount of bias is negligible unless n is very
  small
• r is essentially unbiased if n gt 25
• Accuracy of r as an estimate of rho depends on n
• r is consistent as n increases, the
  difference between r and rho decreases

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Pearson Product-Moment

• Measurement error in X and Y can reduce the value
  of r
• Variance in the sample influences r The greater
  the variability among the observations, the
  greater the value of r
• Shape of distribution influences r
• r can equal 1.0 only when the frequency
  distributions have the same shape
• The less similar the shapes of distribution, the
  lower the maximum value of r

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What if?

• Nominal and interval data
• Point-biserial correlation coefficient

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What if?

• Ordinal and interval data
• Convert interval scores to ranks and calculate
  using Spearman or Kendall statistics

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Measures of Relationships
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Describing Measures of Association -- Davis
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Describing Measures of Association -- Cohen
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Contingency Tables